the question of the triviality of \(u\)  under the above condition was first solved in [1]}; this triviality was also established later in [2]} provided that the velocity \(u\)  is just bounded. Those two results rely heavily on the fact that one has a nice equation for the vorticity \(w := \partial _1 u_{2} - \partial _2 u_{1}\) , i.e.,
\(- \Delta w + u\cdot \nabla w = 0.\) 
